England now leads a delicately poised four-match series 2-1, but several controversial incidents have pockmarked the contest.
Joe Root & Co.at the end of the day's play were 192 for four.
South Africa were tame with the ball early on - Anrich Nortje's bouncer that struck Crawley flush on the helmet a rare moment of aggression - and visibly lacked the spark and speed of the suspended Kagiso Rabada.
Crawley and Sibley completed England's first 100-run opening stand since Cook and Keaton Jennings put on 103 against India in Chennai in December 2016 - but the partnership should have been broken on 41.
Vernon Philander then ensured he will end his final Test with a wicket as Crawley presented a simple catch to Rassie van der Dussen, who made another grab at slip to end a hard 35-ball spell at the crease for Joe Denly (27).
Pieter Malan then spilled a hard diving one-handed chance at cover that would have removed Denly for four and the same batsman was also grassed by Dwaine Pretorius on 25 at short midwicket before his febrile innings ended when he snicked Dane Paterson to slip.
Crawley was undone by his own indecision as he played at a Philander delivery and top edged it to Rassie van der Dussen, who ended up claiming three catches at slip. The Proteas speedster celebrated in an overly aggressive manner after taking the wicket of Joe Root during the third Test in Port Elizabeth.
Off-spinner Dom Bess was left out by England, despite his five-wicket haul in Port Elizabeth last week, with seamer Chris Woakes handed his first start of the series in the visitors' only change.
England hero Ben Stokes could be in hot water with the cricket authorities after he was caught on camera appearing to call a fan a "f*cking four-eyed c*nt" after being dismissed against South Africa.